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summary information for fracture strengths (MPa) of \(n=169\) ceramic bars fired in a particular kiln: \(\bar{x}=89.10\), \(s=3.73\). a. Calculate a (two-sided) confidence interval for true average fracture strength using a confidence level of \(95 \%\). Does it appear that true average fracture strength has been precisely estimated? b. Suppose the investigators had believed a priori that the population standard deviation was about \(4 \mathrm{MPa}\). Based on this supposition, how large a sample would have been required to estimate \(\mu\) to within . \(5 \mathrm{MPa}\) with \(95 \%\) confidence? 15\. Determine the confidence level for each of the following large-sample one-sided confidence bounds: a. Upper bound: \(\bar{x}+.84 s / \sqrt{n}\) b. Lower bound: \(\bar{x}-2.05 s / \sqrt{n}\) c. Upper bound: \(\bar{x}+.67 s / \sqrt{n}\)

Step by step solution

01

Determine the Critical Value for the 95% Confidence Interval

For a 95% confidence interval and a large sample size (n = 169), we use the standard normal distribution (z-distribution). The critical value for a 95% confidence interval is approximately 1.96.

02

Calculate the Margin of Error for the Confidence Interval

The margin of error (ME) is calculated using the formula:\[ME = z \cdot \frac{s}{\sqrt{n}}\]Substitute the given values:\[ME = 1.96 \cdot \frac{3.73}{\sqrt{169}} = 1.96 \cdot 0.286 = 0.56096\]

03

Construct the 95% Confidence Interval for the Mean

With the mean (\(\bar{x}\)) of 89.10 and the margin of error (ME) from Step 2, the confidence interval is:\[89.10 \pm 0.56096 = (88.53904, 89.66096)\]The true average fracture strength appears to be estimated quite precisely.

04

Calculate Sample Size for Desired Margin of Error

To achieve a margin of error of 0.5 MPa, where the population standard deviation \((\sigma)\) is assumed to be 4 MPa, use the sample size formula:\[n = \left( \frac{z \cdot \sigma}{\text{ME}} \right)^2 = \left( \frac{1.96 \cdot 4}{0.5} \right)^2 = \left( \frac{7.84}{0.5} \right)^2 \]\[n = 246.016\]Therefore, a sample of 247 would be necessary for the desired precision.

05

Calculate Confidence Levels for Given Bounds

The confidence level for a one-sided confidence bound is determined based on the z-score used.**a.** Upper bound: \(\bar{x} + 0.84 \cdot \frac{s}{\sqrt{n}}\)- \(z = 0.84\), corresponding to a confidence level of 79.99%**b.** Lower bound: \(\bar{x} - 2.05 \cdot \frac{s}{\sqrt{n}}\)- \(z = 2.05\), corresponding to a confidence level of 97.98%**c.** Upper bound: \(\bar{x} + 0.67 \cdot \frac{s}{\sqrt{n}}\)- \(z = 0.67\), corresponding to a confidence level of 74.86%

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Size Calculation

Understanding how to calculate the appropriate sample size is crucial in research. Sample size influences the reliability of your confidence intervals. If you have a prior belief about the population standard deviation and a desired margin of error, you can compute the sample size needed. Use the formula:

• \[ n = \left( \frac{z \cdot \sigma}{\text{ME}} \right)^2 \]

where \( n \) is the sample size, \( z \) is the z-score for your confidence level, \( \sigma \) is the population standard deviation, and ME is the margin of error. For example, with a z-score of 1.96, population standard deviation of 4, and a margin of error of 0.5 MPa, you'd calculate:

• \[ n = \left( \frac{1.96 \cdot 4}{0.5} \right)^2 = 247 \]

This tells you that a sample of 247 is needed for 95% confidence. Larger sample sizes generally result in more precise estimates of the population parameters.

Mean Estimation

Estimating the mean of a population is fundamental in statistics. The sample mean (\(\bar{x}\)) gives us an estimate of the population mean (\(\mu\)). In practical terms, the sample mean is the arithmetic average of the observed values in your sample data. This is calculated using:

• \[ \bar{x} = \frac{\sum x_i}{n} \]

where \( \sum x_i \) is the sum of all observed values \( x \), and \( n \) is the total number of observations. For example, if the mean fracture strength is given as 89.10 MPa, this is an estimate of the true mean fracture strength of the population. Estimations like this are necessary for making inferences about larger groups based on a sample.

Margin of Error

The margin of error (ME) represents the amount of random sampling error in the estimation of a population parameter. It indicates the range in which we can expect the true population parameter to lie with a certain level of confidence. To calculate the margin of error, use:

• \[ ME = z \cdot \frac{s}{\sqrt{n}} \]

where \( z \) is the z-score associated with your desired level of confidence, \( s \) is the sample standard deviation, and \( n \) is the sample size. For the example of the ceramic bars, we calculated:

• \[ ME = 1.96 \cdot \frac{3.73}{\sqrt{169}} = 0.56096 \]

This margin of error tells us by how much the sample mean can be expected to differ from the population mean.

Z-distribution

In statistics, the z-distribution is a key tool used to determine how far a sample mean is from the population mean when the population standard deviation is known. It assumes a normal distribution with a mean of 0 and a standard deviation of 1. The z-distribution is often used when calculating confidence intervals for large sample sizes (typically \( n \geq 30 \)). The reliability of your estimation depends on the z-value obtained from this distribution. For a 95% confidence interval, we use a critical z-value around 1.96. This z-value signifies that there is a 95% probability that the sample mean falls within 1.96 standard deviations of the true mean. Understanding and applying the z-distribution is vital for accurately estimating population parameters from sample data.

Confidence Level Determination

Determining the confidence level is essential for understanding how reliable an estimate of a population parameter is. The confidence level, usually expressed as a percentage, indicates the probability that the confidence interval contains the true population parameter. To establish this, we use the z-score associated with the confidence level. For instance, if you arrive at a z-score of 1.96, it corresponds to a 95% confidence level. This means we are 95% confident that the true mean lies within our calculated interval. Specific z-scores correspond to different confidence levels.

• Example: A z-score of 0.84 corresponds to a confidence level of roughly 80% for a one-sided bound.

Recognizing the z-score and its associated confidence level is fundamental in statistical analysis.